![]() Of the potato at time t is equal to three.If you are wondering about the format of the AP Calculus BC test, then you’ve come to the right place! In this short article we will discuss the structure of the exam and kinds of questions you might expect to see on test day. Is our approximation using that equation of the tangent line of the internal temperature So this right over here is the equation for the line tangent to the graph of H at t is equal to zero, and this right over here So then we would get, let's see, negative 16 times three plus 91 is equal to, this is negative 48 plus 91 is equal to, what is that? 43, so this is equal to 43 degrees Celsius. We want to approximate the temperature that this model describes, right over here, but we're So let's say that this is time t equals three right over here. Temperature of the potato at time t equals three. Want to use this equation to approximate the internal It is, do a mini drum roll here, y is equal to negative 16 t plus 91. So just like that, we have the equation for the line tangent to the graph of H at t is equal to zero. What is our initial temperature minus 27? This is, of course, 91 degrees. So if we want to think about H prime of zero, that's going to be equal to negative 1/4 times H of zero minus 27. ![]() Of H with respect to t at time t equals zero, rightĪt this point right over here? Well, we just have to look at this. Graph right at that point, that point zero comma 91, plus 91. Where does it intersect the y-axis here? Well, when t is equal to zero, the value of thisĮquation is going to be 91 because it intersects our Well, it would be theĭerivative of our function at that point, so dH/dt, times t, plus our y-intercept. Of the form y is equal to the slope of the equation We want the equation of the tangent line, which might look something like this, at t equals zero. So what are we going to do? Well, we're gonna think about what's going on at time t equal zero, right over here. The internal temperature of the potato at time t equals three. Line tangent to the graph of H at t equals zero. ![]() Less of a difference, the rate of change, you could imagine, becomes less and less and less negative, as we asymptote towards When there's a big difference, you expect a steeper rate of change. That your rate of change is proportional to the differenceīetween the temperature of the potato and theĪmbient room temperature. Have a negative rate of change, which makes sense. So this part here is going to be positive, but then you multiply T greater than zero, this is going to be a negative value because our potato is greater than 27 for t greater than zero. Is consistent with that, where the rate of change, notice this is for all So this is what you would expect to see, and this differential equation This and then asymptote towards a temperature So you would expect it to look something like Intuitively is that it would start to cool and when its temperature, when there's a big difference between the potato and the room, maybe its rate of change is steeper than when there's a little difference. And so we know at t equals zero, our potato is at 91 degrees. That's what the room temperature is doing. Is 27 degrees Celsius, so I'll just draw there. (a), let's just make sense of what this differentialĮquation is telling us, and let's see if it'sĬonsistent with our intuition. In degrees Celsius and H of zero is equal to 91. Our internal temperature and the ambient room temperature, where H of t is measured To negative 1/4 times our, the difference between Modeled by the function H that satisfies theĭifferential equation dH/dt, the derivative of our internal temperature with respect to time, is equal The internal temperature of the potato at time t minutes can be Little bit greater than that, as t gets larger and larger. ![]() Temperature would approach this, but it would always stay a I would guess that theĪmbient room temperature is 27 degrees Celsius. ![]() Of the potato is greater than 27 degrees for all times The internal temperature of the potato is 91 degrees CelsiusĪt time t equals zero, and the internal temperature At time t equals zero, a boiled potato is takenįrom a pot on a stove and left to cool in a kitchen. Going to cover the famous or perhaps infamous potato problem from the 2017 AP Calculus exam. ![]()
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